CHAOS CONTROL INFINITY HOW TO

To overcome this limitation, the NT is combined with a global convergent method named stability transformation method (STM) to improve. Further, the extended algorithms based on Newton method including relaxed Newton method, parameter adjustment hybrid method etc. The most popular algorithm for detecting UPOs is Newton method (NT), which has the second order convergence speed, but it is heavily dependent on the initial condition, as well as on the behavior of the partial derivatives. Owing to the inherent unstable nature of UPOs, the analytic expressions for every periodic orbit cannot be obtained, only if the chaotic system is a simple nonlinear polynomial with low degree and the period of concern is low. However, detecting the UPOs of nonlinear systems with both numerical calculation and experimental implementation is one of the most challenging problems for nonlinear dynamics. Therefore, it is necessary to detect the UPOs of chaotic system in advance.

CHAOS CONTROL INFINITY HOW TO

A key issue for chaos control is how to determine the control variable varying with time in terms of the past or present state, and incorporate it into a feedback term to achieve the purpose of controlling unstable periodic orbits (UPOs). Chaos control can catch the desired fixed points or periodic orbits embedded in the chaotic attractor of nonlinear dynamical system through implementing the target guidance and position. Due to its theoretical significance and engineering applicability, the chaos control has been paid extensive attention in diverse fields. Generally, there are two complementary attributes to define chaos in both the temporal and spatial aspects: exponentially sensitive dependence on initial conditions, and pattern structure of strange attractors. Keywords: chaotic mapping system, modified stability transformation method, relaxes parameter, chaos control.Ĭhaos is an intrinsically complex phenomenon for deterministic nonlinear dynamical system. Finally, numerical analysis verifies the high efficiency of modified STM proposed in this paper. It is indicated that the number of iterations, the absolute and relative errors between the convergent value and analytical fixed point for combined STM-NT decrease enormously comparing with that of traditional STM.

Moreover, STM is combined with Newton method (NT) to overcome the disadvantage of slow convergence around the static bifurcation point without requiring a priori information of fixed point. Then, the critical parameter that satisfies convergence condition and the optimal parameter corresponding to best performance for STM are determined respectively.

Firstly, according to the information of the fixed point, the stability matrix C is derived, demonstrating that it is unnecessarily an involutory matrix for STM that can also stabilize chaotic system to the fixed point. For the multi-dimensional chaotic mapping system with various forms of bifurcations, it is found that the spectral radius around the static bifurcation point is approximately equal to one and the convergence speed for traditional stability transformation method (STM) which is used to stabilize the unstable fixed points is fairly slow In this paper, a modified STM is proposed to efficiently stabilize a 3D chaotic mapping system to stable fixed points.

Chaos control infinity